November 8, 1901.] 



SCIENCE. 



711 



etry, which are called the parabolic, the 

 hyperbolic and the elliptic geometries. 

 They are also called the Geometries of Eu- 

 clid, of Lobachevski, and of Kiemann." 

 Now Saccheri in his demonstration of Pi'op. 

 XI. makes, almost in the words of Archi- 

 medes, an assumption, introduced by the 

 words ' it is manifest,' which we now call, 

 for convenience, Archimedes' Axiom. In 

 his futile attempts at demonstrating the 

 parallel-postulate, Legendre set forth two 

 theorems, called Legendre's theorems on 

 the angle-sum in a triangle. They are : 



1. In a triangle the sum ofthe three angles 

 can rtever be greater than two right angles. 



2. If in any triangle the sum of the three 

 angles is equal to two right angles, so is it 

 in every triangle. 



In addition to assuming the infinity or 

 two-sidedness of the straight, in his proofs 

 of these theorems Legendre uses essentially 

 the Archimedes Axiom. Thence he gets 

 that the angle-sum in a triangle equaling 

 two right angles is equivalent to the par- 

 allel-postulate, all of which is really what 

 Saccheri gave a century before him, now 

 just reproduced by Barbarin and Manning, 

 as before by De Tilly. Even Hilbert in his 

 ' Yorlesung ueber Euklidische Geometric ' 

 (winter semester, 1898-99), for a chance to 

 see Dr. von Schafer's Authographie of 

 which I am deeply grateful to Professor 

 Bosworth, gives the following five theorems 

 and then says : *' Finally we remark, that 

 it seems as if each of these five theorems 

 could serve precisely as equivalent of the 

 Parallel Axiom.^' They are 



1. The sum of the angles of a triangle is 

 always equal to two right angles. 



2. If two parallels are cut by a third 

 straight, then the opposite (corresponding) 

 angles are equal. 



3. Two straights, which are parallel to a 

 third, are parallel to one another. 



4. Through every point within an angle 

 less than a straight angle, I can always 



draw straights which cut both sides [not 

 perhaps their prolongations]. 



5. All points of a straight have from a 

 parallel the same distance. 



But since then a wonderful discovery has 

 been made by M. Dehn. 



It was known that Euclid's geometry 

 could be built up without the Archimedes 

 axiom. So arises the weighty question : 

 In such a geometry do the Legendre theorems 

 necessarily hold good f 



In other words : Can we prove the 

 Legendre theorems without making use of 

 the Archimedes axiom ? 



This is the question which, at the insti- 

 gation of Hilbert, was taken up by Dehn. 



His article was published July 10, 1900 

 (3fathematische Annalen, 53 Band, pp. 404- 

 439). 



Dehn was able to demonstrate Legendre's 

 second theorem without using any postu- 

 late of continuity, a remarkable advance 

 over Saccheri, Legendre, De Tilly. 



But his second result is far more remark- 

 able, namely, that Legendre's first theorem 

 is indemonstrable without the Arctimedes 

 axiom. 



To prove this startling position, Dehn 

 constructs a new non-Euclidean geometry, 

 which he calls a ' non-Legendrean ' geome- 

 try, in which through every point an in- 

 finity of parallels to any straight can be 

 drawn, yet in which nevertheless the angle 

 sum in every triangle is greater than two 

 right angles. 



Thereby is the undemonstrability of the 

 first Legendre theorem without the help of 

 the Archimedes axiom proven. 



Dehn then discusses the connection be- 

 tween the three different hypotheses rela- 

 tive to the sum of the angles [the three hy- 

 potheses of Saccheri, Barbarin, Manning] 

 and the three different hypotheses relative 

 to the number and existence of parallels. 



He reaches the following remarkable 

 propositions : 



